Having a little better view in person, I don't think it was a Flicker.
Robins are common here, though I haven't seen one in a few months. I didn't get a Robin impression from it.
Looking at some videos of Loggerhead Shrikes from similar distances, they really seem to match what I was seeing, but I'm still interested in other theories. And never having knowingly seen one in person before sort of explains why I was so mystified.